# Puzzle – Walking stick

February 4, 2007 am28 3:54 am

Another gem from Bertie Taylor:

I accidentally dropped my straight walking stick into a buzz saw, which neatly cut it into two pieces. I was so angry I picked up one of the pieces at random and threw it at the saw. It cut that piece into two. What is the probability that I could make a triangle with the three pieces?

12 Comments
leave one →

My cat says the answer is

(1-2x) /4 where x represents the ratio of the shorter piece to the entire length of the stick.

I can’t quite wrap my fingers around the algebra to evaluate your expression — I will keep trying, but

1) we are looking for a number, and

2) when I try some known values, (1-2x)/4 returns probabilities that are low. (imagine the best possible first cut…)

You link to a making-sweets blog. Can you really do that stuff? I can make food, but in my family only my sister bakes. I wish I could.

I don’t remember anything about math, but here’s what I’m thinking:

Either you cut the stick into two different sized pieces, or two same-sized pieces. If the pieces are the same size, they won’t make an end-to-end-to-end triangle, they’ll just like flat. So that won’t work.

If, on the other hand, the pieces are uneven, and you cut the longer piece in half, you have a triangle. If you cut the shorter piece, you don’t (assuming, again, that you mean a triangle where everything has to touch on the edges, not in the middle or something).

I don’t know how that works out mathematically, though. This is the sort of thing I’d write on the extra-credit problems in math class and get back a “???” from my teacher. Um, 25%? Except I think it’s unlikely you’d cut it perfectly in half in the first place.

Honestly, very nice qualitative analysis. One question we will have to deal with is, how long is the longer piece? If it is very long, then we are unlikely to cut it in a way that will leave us with a triangle. If it is just over half the original, we are almost guaranteed to get a triangle.

25% sounds like a reasonable guess. There are a few regular puzzle-solvers here who will tell us just how reasonable it was.

I’m thinking and typing at the same time, so no guarantees that there isn’t some major flaw, but here goes…

The probability that you can form a triangle is equal to the probability that none of the three pieces is greater than half the length of the original stick (which I’ll call L).

For this condition to be satisfied, the piece of the stick you pick up to cut again has to be the longer of the two pieces that result from the first cut AND the cut made in that piece has to leave no piece longer than L/2.

The probability of the 1st condition being met is 0.5.

The probablity of the 2nd condition being met depends on where the first cut was. If the first cut left a shorter piece of length x and a longer piece of length L-x, the cut in the longer piece must be a distance greater than L/2 – x from either end. There’s a region of length x is the middle of the longer piece where this condition is statisfied.

So for a given first cut at x, the probability of picking up the longer stick and making a suitable second cut is 0.5 * x/(L-x).

To get the total probablity, you then need to integrate this from x=0 to x=L/2, for which I get 0.5*(-0.5-ln(0.5))=0.0965735903, which is smaller than I would have expected.

Note that the x=L/2 case is a bit quirky, because in that case it doesn’t matter which stick you pick up — you either will or won’t be able to make a triangle, depending on whether you consider a 0,0,180 figure a triangle.

Cool problem. I get the same answer as rdt, though I had to actually work out the second probability number by explicitly writing out the triangle inequalities for any fraction of the bigger piece. True to my nom de clavier, it is still not clear to me how rdt could just get it by intuition.

The quirky x=L/2 case is completely irrelevant to me since it has probability 0.

Clueless.

rdt and clueless made a small error: because x is only varying from 0 to 1/2 (not to 1), you need to divide by 1/2 (the integral of 1 with respect to x as x varies from 0 to 1/2). The clue here is that rdt’s integral depends on L. So the actual answer is twice their value, or ln(2) – 1/2.

25% is the answer to a different, but very similar, question: if break my walking stick into three pieces at two randomly selected points, what is the probability that the three pieces form a triangle?

A small error!!! I was a 100% off.

Anyway, just to convince myself, I ran a small simulation for the problem and arrived at the same answer (ln(2)-0.5)

Ok… intuitively, 0.09657 seemed small. It took me a while to understand the factor of two, but it finally sunk in (and Clueless saved me the trouble of writing the simulation…).

— Rachel

Yeah, but the trouble with simulations is that you can usually get what you want!

And teachingsmarter’s 25% was off, but not dramatically. 19.3%

Using n discrete intervals, we can get to almost 18% with 20 intervals.

My simulation came up with 18.7% +/- 0.4% in 10000 trials

— Rachel